Measuring distributed polarization crosstalk in polarization maintaining fiber and optical birefringent material

BACKGROUND

This patent document relates to devices, systems and techniques for measuring optical polarization property in optical materials and devices.

Optical polarization is an important parameter of an optical signal in various optical devices, systems and applications. The optical polarization of an optical signal can change or can be altered by interacting with an optical medium having optical birefringence in which light experiences different refractive indices at different optical polarizations. Fibers, for example, may be optically birefringent and light propagating in such fibers can change its polarization. The birefringence of a fiber may change with time, often randomly with the fluctuations in the operating conditions such as stresses or temperatures in the fiber.

Polarization maintaining (PM) fiber has high birefringence and supports two discrete polarization modes, HE^Slow₁₁and HE^fast₁₁, that are along mutually orthogonal slow and fast axes of the PM fiber. The refractive index of the PM fiber for light polarized along the slow axis in the mode HE^Slow₁₁is higher than the refractive index of the PM fiber for light polarized along the fast axis in the mode HE^fast₁₁. When the light coupled into the PM fiber is linearly polarized along the slow axis of the PM fiber, only HE^Slow₁₁mode is excited and the optical polarization of the guided light is maintained along the slow axis; conversely, when the light coupled into the PM fiber is linearly polarized along the fast axis of the PM fiber, only HE^fast₁₁mode is excited and the optical polarization of the guided light is maintained along the fast axis. This characteristics of preserving optical polarization in the PM fiber can be used in various applications, such as fiber optic gyroscopes, integrated optics devices, high-performance interferometer and Polari metric sensors, quantum key distribution, and fiber lasers. Perturbations to PM fiber, such as stresses exerted on PM fiber, may cause optical coupling or crosstalk between the two orthogonal polarization modes where optical energy of one polarization mode transfers to optical energy of another polarization mode or vice versa.

SUMMARY

This document includes techniques and devices for measuring the distribution of polarization crosstalk in birefringence optical media including polarization maintaining fiber based on suppressing the number and magnitude of ghost interference peaks.

In one aspect, a method is provided for measuring distribution of polarization coupling in an optical birefringent medium. This method includes coupling a linearly polarized light of a broadband spectrum into an optical birefringent medium in a direction along which the optical birefringent medium supports two orthogonal polarization modes due to optical birefringence to produce an optical output signal out of the optical birefringent medium; directing the optical output signal out of the optical birefringent medium into an optical delay device which causes an added optical delay greater than an optical delay between the two orthogonal polarization modes caused by the optical birefringent medium in the optical output signal to produce a modified optical output signal; directing modified optical output signal to transmit through a linear optical polarizer which is polarized in a direction to cause a mixing between the two orthogonal polarization modes in optical transmission light of the linear optical polarizer; directing the optical transmission light of the linear optical polarizer into an optical interferometer to obtain optical interference of light between the two orthogonal polarization modes in the optical birefringent medium; and processing the obtained optical interference to identify locations of the polarization coupling between the two orthogonal polarization modes in the optical birefringent medium.

In another aspect, a device is provided for measuring distribution of polarization coupling in an optical birefringent medium. This device includes a mechanism that couples a linearly polarized light of a broadband spectrum into an optical birefringent medium in a direction along which the optical birefringent medium supports two orthogonal polarization modes due to optical birefringence to produce an optical output signal out of the optical birefringent medium; and an optical delay device located downstream from the optical birefringent medium in an optical path of the optical output signal out of the optical birefringent medium to produce a modified optical output signal. The optical delay device is structured to cause an added optical delay greater than an optical delay between the two orthogonal polarization modes caused by the optical birefringent medium in the optical output signal to produce the modified optical output signal. This device includes a linear optical polarizer which is polarized in a direction to cause a mixing between the two orthogonal polarization modes in optical transmission light of the linear optical polarizer, the linear optical polarizer placed in an optical path of the modified optical output signal to produce the optical transmission light; an optical interferometer located to receive the optical transmission light of the linear optical polarizer and structured to obtain optical interference of light between the two orthogonal polarization modes in the optical birefringent medium; and a processing device that processes the obtained optical interference to identify locations of the polarization coupling between the two orthogonal polarization modes in the optical birefringent medium. The processing device can also be configured to process the obtained optical interference to obtain magnitudes of the polarization coupling at the identified locations.

In another aspect, a device is provided for measuring distribution of polarization coupling in an optical birefringent medium and includes a light source that produces a light beam of a broad spectral band; a mechanism that couples light of the light beam in a linear polarization into an optical birefringent medium in a direction along which the optical birefringent medium supports two orthogonal polarization modes due to optical birefringence to produce an optical output signal out of the optical birefringent medium; a linear optical polarizer which is polarized in a direction to cause a mixing between the two orthogonal polarization modes in optical transmission light of the linear optical polarizer, the linear optical polarizer placed in an optical path of the optical output signal of the optical birefringent medium to produce the optical transmission light; an optical interferometer made of fiber components and located to receive the optical transmission light of the linear optical polarizer and structured to obtain optical interference of light between the two orthogonal polarization modes in the optical birefringent medium; an optical detector that receives optical output of the optical interferometer and produces a detector output having information of the obtained optical interference; and a processing device that processes the obtained optical interference to identify locations of the polarization coupling between the two orthogonal polarization modes in the optical birefringent medium.

In another aspect, an apparatus described includes a broadband light source polarized linearly along the axis of PM fiber under test; an optical input port to receive input optical signal from the output of PM fiber under test; a delay device to generate additional optical delay between two orthogonal polarization axis of input port at output of this device; a polarizer which mixes optical signals with orthogonal polarization states after the said delay device to the same polarization along the transmission direction of polarizer; an interferometer which receives the optical signal after said polarizer and generates interference signal; a control and processing device which receives the output signal and gives the position and amplitude information of crosstalk coupling points along PM fiber.

An exemplary of the delay device includes a polarization beam splitter (PBS) which receives optical signal from fiber or free space, and splits the received optical beam into to a first beam and a second beam polarized orthogonal; a polarization beam combiner (PBC) combining the said first beam and the said second beam into a light beam at output of said PBC. The optical path difference traveled by the first beam and second beam generates an additional delay between two orthogonal polarizations at output.

Another exemplary of the delay device composes includes a polarization beam splitter (PBS) which receives optical signal from fiber or free space, then splits the received optical beam into a first beam and a second beam polarized orthogonal. After passing a half wave plane or 45-degree rotator, the first beam is reflected back to PBS's port where the first beam comes. After passing through a half wave plane or 45-degree rotator, the second beam is reflected back to the PBS port where the second beam comes. Then the first and second beam are combined in the PBS and output from the forth port of PBS.

The optical path difference traveled by the first and second beam generates an additional delay between two orthogonal polarizations at output. The additional delay added between orthogonal polarization of PM fiber under test can effectively eliminate the ghosts peaks for measuring distributed crosstalk in PM fiber.

These and other aspects and their implementations area described in details in the description, the drawings and the claims.

BRIEF DESCRIPTION OF DRAWINGS

<figref idrefs="DRAWINGS">FIG. 1</figref> shows an exemplary device for measuring spatial distribution of polarization crosstalk along a PM fiber by using an optical interferometer, where <figref idrefs="DRAWINGS">FIG. 1A</figref> shows the components of the device, <figref idrefs="DRAWINGS">FIG. 1B</figref> illustrates the orientation of the optical polarizer with respect to optical axes of the PM fiber and <figref idrefs="DRAWINGS">FIG. 1C</figref> illustrates a situation where stresses are present at multiple locations along the PM fiber to induce cross talk between the two orthogonal polarization modes of the PM fiber.

<figref idrefs="DRAWINGS">FIG. 2A</figref> shows an exemplary device for measuring spatial distribution of polarization crosstalk along a PM fiber by providing an optical delay device between the PM fiber under test and the optical interferometer, where inserts further illustrate operation of the device.

<figref idrefs="DRAWINGS">FIG. 2B</figref> illustrates from the top to the bottom, a PM fiber under stress at three coupling points along the PM fiber and the wave packet sequences polarized along the slow and fast axes at output of the PM fiber, wave packet sequence in the device in <figref idrefs="DRAWINGS">FIG. 1</figref>, and wave packet sequence in the device in <figref idrefs="DRAWINGS">FIG. 2A</figref> with the optical delay device.

<figref idrefs="DRAWINGS">FIG. 3</figref> shows an example of an optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b>.

<figref idrefs="DRAWINGS">FIG. 4</figref> shows another example of an optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b>.

<figref idrefs="DRAWINGS">FIG. 5</figref> shows another example of an optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b>.

<figref idrefs="DRAWINGS">FIG. 6</figref> shows another example of an optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b> using only one polarization beam splitter (PBS) cube.

<figref idrefs="DRAWINGS">FIG. 7</figref> shows another example of an optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b> using only one PBS cube.

<figref idrefs="DRAWINGS">FIG. 8</figref> shows yet another example of an optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b> using one four-ports pigtailed PBS.

<figref idrefs="DRAWINGS">FIG. 9</figref> shows an exemplary apparatus for measuring distributed polarization crosstalk in PM fiber.

<figref idrefs="DRAWINGS">FIG. 10</figref> shows another exemplary apparatus for measuring distributed polarization crosstalk in PM fiber using fiber pigtailed components.

<figref idrefs="DRAWINGS">FIG. 11</figref> shows an example of a device for measuring the attenuation ratio between two orthogonal polarization modes of light guided in a waveguide device.

<figref idrefs="DRAWINGS">FIG. 12</figref> shows an example of a process for measuring distributed polarization crosstalk in an optical birefringent medium such as PM fiber.

DETAILED DESCRIPTION

Examples for implementing techniques and devices for measuring distribution of crosstalk between two polarization modes along PM fiber are provided based on optical interferometric measurements on output light from the PM fiber. The described techniques and devices can be used to effectively suppress the number and magnitude of ghost peaks generated by multiple coupling points or locations where stresses are applied to the PM fiber to cause the crosstalk and the optical interference among these multiple coupling points along the PM fiber. Broadband light can be used in the described techniques and devices to obtain spatially resolved distribution of stresses along the PM fiber by analyzing stress-induced polarization cross-coupling along the length of the PM fiber. High measurement sensitivity, a wide dynamic range, and high spatial measurement accuracy can be achieved by using the described techniques and devices.

In a PM fiber, when the launched light is perfectly aligned along slow or fast axis at the input of the PM fiber, the optical coupling between the two polarization modes in the PM fiber occurs because intrinsic defects exist in the PM fiber or/and external stresses exerted on the PM fiber. The mode coupling between the slow axis and fast axis of the PM fiber can be characterized with polarization crosstalk. One way to represent the polarization crosstalk is the light intensity ratio between the light in the two polarization modes with optical polarizations along the slow and fast axes, respectively. In practical applications, it is desirable to identify the position of the polarization crosstalk in the PM fiber and to measure the degree of the polarization crosstalk. For example, in fiber optic gyroscopes application, the polarization crosstalk measurements can be used to screen the PM fiber before winding PM coil and to control crosstalk degradation during coil winding and to diagnose the PM coil problem after winding. The PM fiber can be used as an optical sensing medium and the polarization crosstalk can be used as a sensing mechanism. For example, the polarization crosstalk measurements can be used to obtain the stress distribution along the PM fiber and monitor space-resolved structural changes along bridges, tunnels, dams, pipeline or pipes for transporting a liquid (e.g., oil) or a gas (e.g., natural gas), or buildings. The polarization crosstalk measurements can also be used to detect an intrusion to a PM fiber link because mechanical disturbances to the PM fiber introduced by the intrusion causes polarization coupling in the PM fiber. The polarization crosstalk measurements can be used for PM fiber quality inspection by identifying defective sections of PM fiber where the crosstalk occurs, enabling the manufacturers or users to remove the defective fiber sections or take preventive measures to mitigate the impact of such defects. The polarization crosstalk measurements can also be used for measuring high polarization extinction ratios of a polarizing waveguide, obtaining the autocorrelation function of a light source, measuring the birefringence of a PM fiber and the lengths of PM fibers and single-mode (SM) fibers, and matching the optical path lengths of an interferometer.

Optical interference between light waves along the slow and fast axes of the PM fiber can generate real optical interference signals generated at the cross coupling locations in the PM fiber and ghost interference signals caused by the multiple coupling of light wave among multiple crosstalk points. The ghost signals can be strong when there are several strong coupling points on PM fiber, and thus result in wrong identification of crosstalk position and amplitude.

<figref idrefs="DRAWINGS">FIG. 1</figref> shows an exemplary device <b>100</b> for measuring spatial distribution of polarization crosstalk along a PM fiber by using an optical interferometer, where <figref idrefs="DRAWINGS">FIG. 1A</figref> shows components of the device, <figref idrefs="DRAWINGS">FIG. 1B</figref> illustrates the orientation of the optical polarizer with respect to optical axes of the PM fiber and <figref idrefs="DRAWINGS">FIG. 1C</figref> illustrates a situation where stresses are present at multiple locations along the PM fiber to induce cross talk between the two orthogonal polarization modes of the PM fiber.

In this example, a broadband light (<b>101</b>) from a broadband light source is directed into the PM fiber at position A (<b>110</b>). The light (<b>101</b>) has one polarization component aligned to the slow axis of the PM fiber. Stress at position B induces polarization coupling between the two orthogonal polarizations along the fast and slow axes of the PM fiber and produces a polarization component aligned to the fast axis. Because the two polarization components travel at different group velocities in the PM fiber, the two polarization components experience a delay difference at the output (<b>111</b>) of the fiber (position C): <br /><i>Δz=n</i>_s<i>z−n</i>_f<i>z=Δnz</i>  (1)<br /> where n_sand n_fare the refractive indices of the slow and fast axes, respectively, the difference between the two refractive indices Δn is the birefringence, and z is the distance between the coupling point B and the output point C. If an optical polarizer (<b>120</b>) with its optical polarization axis oriented at 45 degrees from the slow axis (<figref idrefs="DRAWINGS">FIG. 1B</figref>) is placed after the fiber output (<b>111</b>), one half of the optical power in each of the two polarization components passes through the polarizer (<b>120</b>) and emerges with the same polarization state which is linear, aligned to the polarizer axis of the polarizer (<b>120</b>).

Therefore, when an optical interferometer is used to receive the output light from the polarizer (<b>120</b>), the presence of the polarizer (<b>120</b>) can cause the received light, which includes two polarization components that are respectively in the two polarization modes in the PM fiber, to optically interfere. This optical interference can then be used to perform the polarization crosstalk measurements.

In <figref idrefs="DRAWINGS">FIG. 1</figref>, a Michelson interferometer is shown as an example for implementing the optical interferometer. A beam splitter <b>130</b> is provided to receive the output light from the polarizer <b>120</b> and splits the received light into a first beam along a first optical path <b>142</b> to a fixed mirror <b>140</b> and a second beam along a second optical path <b>143</b> to a movable mirror <b>141</b>. An actuator is engaged to the movable mirror <b>141</b> to move the position of the movable mirror <b>141</b> to adjust the optical path length of the second optical path <b>143</b> relative to the first optical path <b>142</b>. The two mirrors <b>140</b> and <b>144</b> reflect the two beams back to retrace the first and second optical paths to reach the beam splitter <b>130</b>. The reflected beams from the two mirrors <b>140</b> and <b>141</b> spatially overlap with each other at the beam splitter <b>130</b> and optically interfere to produce the optical output <b>132</b> which contains the optical interference signal which has periodic interference peaks as the mirror <b>141</b> is moved in position. The distance associated with the movement of the mirror <b>141</b> between the two adjacent interference peaks in the optical interference signal is Δnz and, accordingly, from Eq. (1), the location of the coupling point in the PM fiber is z=Δz/Δn. The coupling point can therefore be located using the interference graph. The coupling ratio can also be calculated from the strength of the interference peaks.

<figref idrefs="DRAWINGS">FIG. 1C</figref> illustrates presence of multiple coupling points in the PM fiber. Under this condition, the measurement process is more complicated. Assuming there are (n+1) coupling points (x₀x₁x₂. . . x_n) in the PM fiber, a linearly-polarized input wave packet (<b>112</b>) along the slow axis splits to 2ⁿsmall wave packets along the slow axis and 2ⁿsmall wave packets along the fast axis at the output end of PM fiber (<b>113</b>). Therefore, after the ith coupling point, the two wave packets sequences P_siand P_fipolarized along the slow axis and fast axis respectively include 2ⁱwave packets in each sequence and their optical paths length can be described as

<maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>P</mi><msub><mi>s</mi><mi>i</mi></msub></msub><mo>=</mo><mrow><mrow><mrow><mo>(</mo><mtable><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>s</mi><mi>i</mi></msub><mo>,</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>s</mi><mi>i</mi></msub><mo>,</mo><mn>2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>s</mi><mi>i</mi></msub><mo>,</mo><mi>j</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>s</mi><mi>i</mi></msub><mo>,</mo><msup><mn>2</mn><mi>i</mi></msup></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo>⁢</mo><msub><mi>P</mi><msub><mi>f</mi><mi>i</mi></msub></msub></mrow><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>f</mi><mi>i</mi></msub><mo>,</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>f</mi><mi>i</mi></msub><mo>,</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>f</mi><mi>i</mi></msub><mo>,</mo><mi>j</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>f</mi><mi>i</mi></msub><mo>,</mo><msup><mn>2</mn><mi>i</mi></msup></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where P_si,j(j=1 to 2ⁱ) and P_{fi, (j=1 to 2}ⁱ) represent the optical patch lengths of the jth wave packet in sequences P_siand P_fi, respectively. The optical path length of the wave packet sequences after the (i+1)th coupling point can be calculated by

<maths id="MATH-US-00002" num="00002"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>P</mi><msub><mi>s</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></msub><mo>=</mo><mrow><msub><mi>P</mi><msub><mi>f</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></msub><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>P</mi><mrow><msub><mi>s</mi><mi>i</mi></msub><mo>⁢</mo><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>P</mi><mrow><msub><mi>s</mi><mi>i</mi></msub><mo>⁢</mo><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>P</mi><mrow><mi>si</mi><mo>,</mo><msup><mn>2</mn><mi>i</mi></msup></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>P</mi><mrow><msub><mi>f</mi><mi>i</mi></msub><mo>⁢</mo><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>P</mi><mrow><msub><mi>f</mi><mi>i</mi></msub><mo>⁢</mo><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>P</mi><mrow><mi>fi</mi><mo>,</mo><msup><mn>2</mn><mi>i</mi></msup></mrow></msub></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

Based on formula (3), the optical path length of the wave packet at output of PM fiber can be obtained by

<maths id="MATH-US-00003" num="00003"><math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><msub><mi>P</mi><mi>sn</mi></msub><mo>=</mo><mi /><mo>⁢</mo><msub><mi>P</mi><mi>fn</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>Ps</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>Ps</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>Ps</mi><mrow><mrow><mi>si</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>Pf</mi><mrow><mi>n</mi><mo>-</mo><mrow><msub><mn>1</mn><mi>i</mi></msub><mo>⁢</mo><mn>1</mn></mrow></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>Pf</mi><mrow><mi>n</mi><mo>-</mo><mrow><msub><mn>1</mn><mi>i</mi></msub><mo>⁢</mo><mn>2</mn></mrow></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>Pf</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></msub></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>Ps</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>Ps</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>Pf</mi><mrow><mi>n</mi><mo>-</mo><mrow><msub><mn>1</mn><mi>i</mi></msub><mo>⁢</mo><mn>1</mn></mrow></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>Pf</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>Pf</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> and the corresponding intensity I_snand I_fnof wave packet sequences P_snand P_fncan be calculated by the following formulae:

<maths id="MATH-US-00004" num="00004"><math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><msub><mi>Is</mi><mi>n</mi></msub><mo>=</mo><mi /><mo>⁢</mo><mrow><mfrac><msub><mi>Is</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow></mfrac><mo>⊕</mo><mrow><mfrac><msub><mi>c</mi><mi>n</mi></msub><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow></mfrac><mo>⁢</mo><msub><mi>If</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>3</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><msup><mn>2</mn><mrow><mi>i</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>*</mo><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub><mo>*</mo><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>3</mn></mrow></msub><mo>*</mo><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></msub><mo>*</mo><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>≈</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>3</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><msup><mn>2</mn><mrow><mi>i</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mrow><msub><mi>If</mi><mi>n</mi></msub><mo>=</mo><mi /><mo>⁢</mo><mrow><mrow><mfrac><msub><mi>c</mi><mi>n</mi></msub><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow></mfrac><mo>⁢</mo><msub><mi>Is</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>⊕</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow></mfrac><mo>⁢</mo><msub><mi>If</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>*</mo><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub><mo>*</mo><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>3</mn></mrow></msub><mo>*</mo><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><msup><mn>2</mn><mrow><mi>i</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></msub><mo>*</mo><mrow><msub><mi>c</mi><mi>n</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>3</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>*</mo><msub><mi>c</mi><mi>n</mi></msub></mrow></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> where c_nis the coupling coefficients at point xn, and can be used to represent a crosstalk parameter defined by <br />Crosstalk=abs(10*log <i>c</i>_n)

After passing through the 45° aligned polarizer (<b>120</b>), the two wave packet sequences P_snand P_fn, originally polarized along the slow axis and fast axis in the PM fiber, will be the mixed into one wave packet sequence polarized along transmission direction of the polarizer (<b>120</b>). The optical path length P and the corresponding optical intensity of the wave packet sequence polarized along transmission direction of the polarizer (<b>120</b>) can be calculated as

<maths id="MATH-US-00005" num="00005"><math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><mi>P</mi><mo>=</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>p</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>p</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>p</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><msub><mi>p</mi><msup><mn>2</mn><mi>n</mi></msup></msub></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>Ps</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mfrac><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>Ps</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></msub></mrow><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>Pf</mi><mrow><mi>n</mi><mo>-</mo><mrow><msub><mn>1</mn><mi>i</mi></msub><mo>⁢</mo><mn>1</mn></mrow></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>Pf</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow><mo>+</mo><msub><mi>Pf</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mn>1</mn></mrow></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><msub><mi>n</mi><mi>f</mi></msub></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><msub><mi>P</mi><msub><mi>s</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></msub></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>f</mi><mrow><mi>n</mi><mo>-</mo></mrow></msub><mo>⁢</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mrow><mi>I</mi><mo>≈</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>3</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mi>Is</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><msup><mn>2</mn><mrow><mi>i</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>1</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>3</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mn>3</mn></mrow></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><msub><mi>If</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>⁢</mo><msub><mo>,</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></msub><mo>⁢</mo><mrow><mo>/</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>c</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi /><mo>⁢</mo><mrow><mo>(</mo><mtable><mtr><mtd><msub><mi>Is</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>If</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>

As the mirror <b>141</b> moves to change its position in the second optical path, any two pulses in wave packet sequence P (see formula 7) can generate an interference signal and the position of interference pattern is determined by the delay difference between these two pulses. There are total 2ⁿ*(2ⁿ−1)/2 peaks that are generated in which there are n interference peaks representing the actual coupling points and the rest of the interference peaks are ghosts peaks. These ghost peaks not only generate fake coupling signals, but also can possibly produce compositions at the true interference peaks associated with the true coupling locations. Therefore, the presence of the ghost peaks degrades the measurement accuracy in measuring the crosstalk distribution and amplitude.

Formulae (7) and (8) show that, the wave packet sequence has two groups, one represented by the top half of Formula (7) and comes from Psn₋₁polarized along the slow-axis when in the PM fiber, and another is represented by the bottom half of Formula (7) and comes from Pf_n-1polarized along the fast axis when in the PM fiber. The positions of interference patterns between any two pulses in the group Ps_n-1have nothing to do with the length of the last PM segment (x_n−x_n-1), and their delay difference are all shorter than the (x_n-1−x0)*Δn. The positions of interference patterns between any two pulses in the group Pf_n-1also has nothing to do with the length of last PM segment (x_n−x_n-1), and their delay difference are all less than the (x_n-1−x0)*Δn. For the interference between top and bottom half of wave packet P, the delay difference between any one wave packets from group of Ps_n-1and Pf_n-1, respectively, is (x_n−x_n-1)Δn+(Ps_{n-1, j}−Pf_{n-1, k}). If the length of the last PM segment x_n−x_n-1is longer than the length of the total length (x_n-1−x₀) of the PM segments from 0 to n−1, the interference peaks will split into two groups at position. One group is generated by the interference between any two wave packets in sequence Ps_n-1or P_sf-1; another group is generated by the interference between one wave packet in sequence Psn−1 and one in P_sf-1respectively. A high value for the extinction ratio (ER) of a PM fiber link generally suggests that the coupling coefficients of c1, c2 . . . ci in the PM fiber link are very small so the pulse P<b>1</b> in formula (7) has a relatively high power. If the wave packets generated by over two times coupling and over three order's interference are ignored, there are only n interference signals in the second interference group and the corresponding delay difference between the first optical path <b>142</b> as the reference arm of the optical interferometer (in <figref idrefs="DRAWINGS">FIG. 1A</figref>) and the second optical path <b>143</b> as the changing arm of the optical interferometer are:

<maths id="MATH-US-00006" num="00006"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>n</mi></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>3</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>n</mi></mrow></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mi>…</mi></mtd></mtr><mtr><mtd><mrow><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><mi>Δ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo>⁢</mo><mi>n</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths><br /> which exactly corresponds to the coupling points from 0 to n−1 at the PM fiber.

To reduce the ghost interference peaks, an optical delay can be inserted between the PM fiber and the polarizer (<b>212</b>) to selectively cause an additional delay in light in one of the two polarization modes of the PM fiber. <figref idrefs="DRAWINGS">FIG. 2A</figref> shows an exemplary device for measuring spatial distribution of polarization crosstalk along a PM fiber by providing an optical delay device between the PM fiber under test and the optical interferometer, where inserts further illustrate operation of the device. The input light (<b>201</b>) is split to two orthogonal sequences wave packets after passing though the PM fiber under test (<b>202</b>) and the two sequences are polarized along the slow-axis and the fast-axis of the PM fiber, respectively. The delay device <b>210</b> adds an additional delay L between these two orthogonal wave packet sequences, and the delay L in vacuum should be longer than Δn*l where Δn is the birefringence of the PM fiber and l is the length of the PM fiber and the additional delay L is added to the light polarized along the slow axis of the PM fiber in this example. After passing the 45 degree aligned polarizer (<b>220</b>), these two sequences of wave packets with the additional delay L are mixed together with the same polarization state defined by the polarizer (<b>220</b>). An optical interferometer <b>230</b> is provided downstream from the polarizer (<b>22</b>) to produce a serial of interference signals at delays between Δn*l and (L−Δn*l), these interference signals only correspond the real signals caused by polarization coupling at coupling locations and ghosts peaks are suppressed or eliminated. A processing device <b>240</b> is provided to receive the output of the optical interferometer <b>230</b> and processes the data in the output to generate the measurements for the locations of coupling points in the PM fiber and the magnitudes of the coupling at the respective coupling points.

<figref idrefs="DRAWINGS">FIG. 2B</figref> illustrates, from the top to the bottom, a PM fiber under stress at three coupling points along the PM fiber and the wave packet sequences polarized along the slow and fast axes at output of the PM fiber, wave packet sequence in the device in <figref idrefs="DRAWINGS">FIG. 1</figref> without the additional optical delay device, and wave packet sequence in the device in <figref idrefs="DRAWINGS">FIG. 2A</figref> with the optical delay device.

Consider a situation where there are three coupling points x<b>1</b>, x<b>2</b> and x<b>3</b> along the PM fiber and the light input to the PM fiber has no fast axis component and is polarized along the slow axis of the PM fiber. At each coupling point, light is coupled not only from the polarization mode along the slow axis to the polarization mode along the fast axis, but also from the polarization mode along the fast axis to the polarization mode along the slow axis. As a result of this coupling, the resulted wave packet series output by the PM fiber include wave packets caused by multiple couplings.

As shown <figref idrefs="DRAWINGS">FIG. 2B</figref>, four wave packets S₀, S₁₂, S₂₃and S₁₃emerging at the output are aligned to the slow axis of the PM fiber. S₀is the light that directly propagates along the slow-axis from input x₀to the end of the fiber x_end. The optical path length traveled by S₀is n_sL where L is the length of the fiber. S₁₂, S₂₃and S₁₃are generated by coupling along the PM fiber. More specifically, S₁₂is the light propagated from input to x₁along the slow axis, then from x₁to x₂along the fast axis, then from x₂to the output along the slow axis; S₂₃is the light propagated from input to x₂along the slow axis, then from x₂to x₃along the fast axis, then from x₃to the output along the slow axis; and S₁₃is the light propagated from input to x₁along the slow axis, then from x₁to x₃along the fast axis, then from x₃to the output along the slow axis.

Along the fast axis, three main packets f₁, f₂, and f₃emerge at the output of the fiber. They are generated by coupling from the slow axis to the fast axis at points x₁, x₂and x₃, respectively. More specifically, the wave packet f₁is the light propagating from input to x₁along the slow axis, then from x₁to output along the fast axis; the wave packet f₂is the light propagating from input to x₁along the slow axis, then from x₂to output along the fast axis; and the wave packet f₃is the light propagating from input to x₃along the slow axis, then from x₃to output of the fast axis. The wave packet generated by a third order coupling from input to x₁along slow axis, from x₁to x₂along fast axis, from x₂to x₃along slow axis, then from x₃to output along fast axis. In many practical situations, this third order coupling can be neglected because its optical power is very low.

After passing through the 45° oriented analyzer, the wave packets aligned to the slow and fast axes will be mixed together (<b>270</b> and <b>280</b> in <figref idrefs="DRAWINGS">FIG. 2B</figref>). If this mixed light is input to an interferometer, a series of interference peaks can be observed as the delay in one arm of the interferometer is changed. Table 1 lists interference peaks for the example shown in <figref idrefs="DRAWINGS">FIG. 2B</figref>. Interference peaks generated by the interference between S₀<img id="CUSTOM-CHARACTER-00001" he="2.12mm" wi="3.89mm" file="US08599385-20131203-P00001.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" />f₁, S₀<img id="CUSTOM-CHARACTER-00002" he="2.12mm" wi="3.89mm" file="US08599385-20131203-P00001.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" />f₂, and S₀<img id="CUSTOM-CHARACTER-00003" he="2.12mm" wi="3.89mm" file="US08599385-20131203-P00001.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" />f₃represent the actual three coupling points x₁, x₂and x₃in the PM fiber and other peaks listed in Table 1 are ghost peaks which can undesirably cause errors in identification of the actual coupling points x₁, x₂and x₃. Ghost peaks can also be superimposed on the real peaks, reducing the crosstalk measurement accuracy.

In order to suppress the number and magnitude of the undesired ghost peaks, the delay device <b>210</b> in <figref idrefs="DRAWINGS">FIG. 2A</figref> can be inserted between the PM fiber's output and the polarizer's input. This delay device is polarization selective and can add an additional delay between the slow axis and the fast axis of the PM fiber. Thus, the two wave packet sequences from the fast-axis and slow-axis are separated in time after the light passes through the analyzer (see <b>280</b> in <figref idrefs="DRAWINGS">FIG. 2B</figref>). If we preset the same delay offset between the fixed and moving arms in the interferometer, the zero order, second order and most higher order interference signals will not be generated as the delay line scans (see Table 1); therefore, most of the ghost peaks disappear during measurement. Consequently, the device in <figref idrefs="DRAWINGS">FIG. 2A</figref> has higher position measurement accuracy, higher dynamic range and higher sensitivity than other interferometer-based devices such as the device in <figref idrefs="DRAWINGS">FIG. 1</figref>.

The polarization-selective optical delay device (<b>210</b>) in <figref idrefs="DRAWINGS">FIG. 2A</figref> can be implemented in various configurations and can be selected based on the needs of a particular application for the device (<b>210</b>) in <figref idrefs="DRAWINGS">FIG. 2A</figref>. Light in the two polarization modes of the PM fiber can be separated into two optical signals along two separate paths by using a polarization beam splitter and a variable optical delay mechanism can be used to cause a variable optical delay between the two separated optical signals before recombining the two separated signals into a combined optical signal for subsequent processing by the downstream linear optical polarizer and the optical interferometer. <figref idrefs="DRAWINGS">FIGS. 3</figref>, <b>4</b>, <b>5</b>, <b>6</b>, <b>7</b> and <b>8</b> show several examples for implementing such an optical delay device. These examples can be configured as fixed optical delay devices that produce a desired optical delay ΔL(&gt;Δn*l where l is the length of PM fiber under test) or a variable delay that can be controlled to be at the above desired optical delay ΔL.

<tables id="TABLE-US-00001" num="00001"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="1"><colspec colname="1" colwidth="217pt" align="center" /><thead><row><entry namest="1" nameend="1" rowsep="1">TABLE 1</entry></row></thead><tbody valign="top"><row><entry namest="1" nameend="1" align="center" rowsep="1" /></row><row><entry>comparison between invented method to previous art</entry></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="3"><colspec colname="1" colwidth="70pt" align="left" /><colspec colname="2" colwidth="77pt" align="left" /><colspec colname="3" colwidth="70pt" align="left" /><tbody valign="top"><row><entry /><entry>Device in FIG. 1</entry><entry>Device in FIG. 2A</entry></row><row><entry namest="1" nameend="3" align="center" rowsep="1" /></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="4"><colspec colname="1" colwidth="14pt" align="left" /><colspec colname="2" colwidth="56pt" align="left" /><colspec colname="3" colwidth="77pt" align="left" /><colspec colname="4" colwidth="70pt" align="left" /><tbody valign="top"><row><entry /><entry>Zero-order</entry><entry>S₀ <img id="CUSTOM-CHARACTER-00004" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> S₀, S₁₂ <img id="CUSTOM-CHARACTER-00005" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> S₁₂,</entry><entry>None</entry></row><row><entry /><entry>interference</entry><entry>S₂₃ <img id="CUSTOM-CHARACTER-00006" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> S₂₃,</entry><entry /></row><row><entry /><entry /><entry>S₁₃ <img id="CUSTOM-CHARACTER-00007" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> S₁₃, f₁ <img id="CUSTOM-CHARACTER-00008" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₁,</entry><entry /></row><row><entry /><entry /><entry>f₂ <img id="CUSTOM-CHARACTER-00009" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂, f₃ <img id="CUSTOM-CHARACTER-00010" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₃</entry><entry /></row><row><entry /><entry>Interference from</entry><entry>S₀ <img id="CUSTOM-CHARACTER-00011" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₁, S₀ <img id="CUSTOM-CHARACTER-00012" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂,</entry><entry>S₀ <img id="CUSTOM-CHARACTER-00013" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₁, S₀ <img id="CUSTOM-CHARACTER-00014" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂,</entry></row><row><entry /><entry>single-order</entry><entry>S₀ <img id="CUSTOM-CHARACTER-00015" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₃</entry><entry>S₀ <img id="CUSTOM-CHARACTER-00016" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₃</entry></row><row><entry /><entry>coupling</entry><entry /><entry /></row><row><entry /><entry>Interference from</entry><entry>S₀ <img id="CUSTOM-CHARACTER-00017" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> S₁₂, S₀ <img id="CUSTOM-CHARACTER-00018" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /></entry><entry>None</entry></row><row><entry /><entry>second order</entry><entry>S₂₃, S₀ <img id="CUSTOM-CHARACTER-00019" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> S₁₃</entry><entry /></row><row><entry /><entry>coupling</entry><entry>f₁ <img id="CUSTOM-CHARACTER-00020" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂, f₁ <img id="CUSTOM-CHARACTER-00021" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₃,</entry><entry /></row><row><entry /><entry /><entry>f₂ <img id="CUSTOM-CHARACTER-00022" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₃</entry><entry /></row><row><entry /><entry>Interference from</entry><entry>S₁₂ <img id="CUSTOM-CHARACTER-00023" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₃, S₁₂ <img id="CUSTOM-CHARACTER-00024" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" />  S₂₃,</entry><entry>S₁₂ <img id="CUSTOM-CHARACTER-00025" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₃, S₁₂ <img id="CUSTOM-CHARACTER-00026" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂,</entry></row><row><entry /><entry>higher order</entry><entry>S₁₂ <img id="CUSTOM-CHARACTER-00027" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> S₁₃,</entry><entry>S₁₂ <img id="CUSTOM-CHARACTER-00028" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₁</entry></row><row><entry /><entry>coupling</entry><entry>S₁₂ <img id="CUSTOM-CHARACTER-00029" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂, S₁₂ <img id="CUSTOM-CHARACTER-00030" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₁</entry><entry>S₂₃ <img id="CUSTOM-CHARACTER-00031" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂, S₂₃ <img id="CUSTOM-CHARACTER-00032" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₁</entry></row><row><entry /><entry /><entry>f₃ <img id="CUSTOM-CHARACTER-00033" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" />  S₂₃, f₃ <img id="CUSTOM-CHARACTER-00034" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" />  S₁₃</entry><entry>S₁₃ <img id="CUSTOM-CHARACTER-00035" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂, S₁₃ <img id="CUSTOM-CHARACTER-00036" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₁</entry></row><row><entry /><entry /><entry>S₂₃ <img id="CUSTOM-CHARACTER-00037" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> S₁₃S₂₃ <img id="CUSTOM-CHARACTER-00038" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂,</entry><entry /></row><row><entry /><entry /><entry>S₂₃ <img id="CUSTOM-CHARACTER-00039" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₁</entry><entry /></row><row><entry /><entry /><entry>S₁₃ <img id="CUSTOM-CHARACTER-00040" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₂, S₁₃ <img id="CUSTOM-CHARACTER-00041" he="1.44mm" wi="2.79mm" file="US08599385-20131203-P00002.TIF" alt="custom character" img-content="character" img-format="tif" orientation="portrait" inline="no" /> f₁</entry></row><row><entry namest="1" nameend="4" align="center" rowsep="1" /></row></tbody></tgroup></table></tables>

<figref idrefs="DRAWINGS">FIG. 3</figref> gives an exemplary of an optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b>. The input orthogonal linear polarization states are aligned to the axis of the polarization beam splitter (<b>310</b>, PBS) at input <b>311</b>, and split into two signals at two output ports <b>312</b> (P<b>1</b>) and <b>313</b> (P<b>2</b>) that are coupled to two separate arms <b>320</b> and <b>321</b>, respectively. An optical delay line is inserted into one of the optical arms <b>320</b> and <b>321</b>, for example, in the arm <b>320</b> as illustrated, to cause the desired optical delay between the two polarizations and the delay can be varied. A polarization beam combiner (PBC <b>330</b>) is coupled to two arms <b>320</b> and <b>321</b> and has input ports <b>332</b> and <b>333</b> to respectively receive these two orthogonal linear polarizations from the two arms <b>320</b> and <b>321</b>. The PBC <b>330</b> combines the received optical signals from the two arms <b>320</b> and <b>321</b> into a combined signal with a delay L between the two polarization states P<b>1</b> an P<b>2</b> at output <b>331</b> of PBC. The delay L is determined by the difference of the delay in the arm <b>320</b> (Delay₃₂₀) and the delay in the arm <b>321</b> (Delay₃₂₁).

<figref idrefs="DRAWINGS">FIG. 4</figref> gives another exemplary optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b> in the PM fiber. At input, the output terminal of the PM fiber under test is connected to a PM collimator <b>401</b> which receives the light and preserves the polarization states of the light received from the PM fiber. The received light at the PM collimator <b>401</b> is coupled into a free-space PBS (<b>410</b>) of the optical delay device. The optical delay device uses in the PBS (<b>410</b>) to split the incoming light into two orthogonal polarization light beams in s-polarization along a first optical path <b>412</b> and in p-polarization along a second, separate optical path <b>413</b>. A second PBS (<b>440</b>) is provided as a polarization beam combiner (PBC) to receive output beams from the two optical paths <b>412</b> and <b>413</b> and to combine the received light in the s-polarization and p-polarization into a combined optical output <b>443</b>. Mirrors <b>420</b> and <b>421</b> may be used to direct the light in the first optical path <b>413</b> to the PBS <b>440</b>. An transmittance optical medium (<b>422</b>) can be inserted in two one of the two optical paths <b>413</b> and <b>412</b> to cause the desired delay L between the two polarization states. The optical medium (<b>422</b>) may be adjusted to change the delay L. In addition, the mirrors <b>420</b> and <b>421</b> may be adjusted to change the delay L.

The collimator <b>401</b> should be aligned to make polarization P<b>1</b> and P<b>2</b> parallel to s-pol or p-pol axis of PBS <b>410</b>, respectively. P-polarization light directly travels to the p-pol input <b>441</b> of a free-space PBC (<b>440</b>) and its optical path can be adjusted by changing the distance between PBS (<b>410</b>) and PBS (<b>440</b>). S-pol. light arrives s-pol input <b>442</b> of PBS (<b>440</b>) after passing the reflector <b>420</b>, the delay medium <b>422</b> and the reflector <b>421</b>. The optical path length of the first optical path <b>413</b> can be adjusted by changing the position of reflectors <b>420</b> and <b>421</b>, or add some transmittance optical medium (<b>422</b>) in the arm <b>432</b>. In PBC (<b>440</b>), the s-pol. and p-pol lights are combined into one light beam at output; the output light beam can be coupled into a PM fiber using a collimator when a fiber-based interferometer is used or directly coupled into a free space interferometer for measurement.

<figref idrefs="DRAWINGS">FIG. 5</figref> gives a fiber-based exemplary optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b>. At input, the output of PM fiber under test is connected into the PM input <b>511</b> of a pigtailed PBS <b>510</b>, the slow axis the PM fiber under test should be aligned to the slow axis of input PM fiber of PBS, thus polarizations P<b>1</b> and P<b>2</b> will be separated into two separated optical signals at the output ports <b>512</b> and <b>513</b> of PBS <b>510</b>. Two fiber loops <b>520</b> and <b>521</b> are coupled to the ports <b>512</b> and <b>512</b> of the PBS <b>510</b>, respectively, to receive the two separated optical signals in polarizations P<b>1</b> and P<b>2</b> and provide two optical paths for the optical signals. A polarization beam combiner (PBC <b>530</b>) is coupled to the two fiber loops <b>520</b> and <b>521</b> at its inputs <b>532</b> and <b>533</b> to receive the optical signals in the two polarization states P<b>1</b> and P<b>2</b>. The received optical signals are combined by the PBC <b>530</b> into the output PM fiber of PBC (<b>530</b>). A variable delay device, such as a fiber stretcher, can be coupled into one of the two fiber loops <b>520</b> and <b>521</b> to cause the variable delay L between the two polarization states P<b>1</b> and P<b>2</b>.

<figref idrefs="DRAWINGS">FIG. 6</figref> shows an exemplary optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b> using only one PBS cube. At input, the PM fiber under test is connected to a PM collimator <b>601</b> to receive the light in the two polarization states of the PM fiber and the received light is coupled into a free-space PBS (<b>610</b>) from an input port <b>611</b>. The collimator is aligned to make polarization P<b>1</b> and P<b>2</b> parallel to s-pol axis or p-pol axis of PBS <b>610</b>, respectively. P-polarized light and S-polarized light received at the input port <b>611</b> are output at two output ports <b>613</b> and <b>612</b> of the PBS <b>610</b> along two separated optical paths. The P-polarized light passes a delay medium <b>640</b> and an optical polarization component <b>651</b> (e.g., a λ/4 wave plate or 45° Faraday rotator), then is reflected back to PBS <b>610</b> by a mirror <b>650</b>. A 45° Faraday rotator rotates polarization light by 45° when the light passes through the Faraday rotator once and, if the transmitted light is reflected back to pass through the Faraday rotator again, a total rotation of 90° is generated in the reflected light. The S-polarized light output by the port <b>612</b> passes a second delay medium <b>620</b> and a second optical polarization component <b>631</b> (e.g., a λ/4 wave plate or 45° Faraday rotator), then is reflected back to PBS <b>610</b> by a mirror <b>630</b>. The PBS <b>610</b> combines the s-pol. and p-pol light into the combined output <b>660</b> at the output port <b>614</b>. The output <b>660</b> can be coupled into a PM fiber using a collimator for subsequent processing by a fiber-based interferometer or directly coupled into a free space interferometer for measurement. The delay between P<b>1</b> and P<b>2</b> at output <b>660</b> can be adjusted by changing the position of mirror <b>630</b> or <b>650</b>, or insert desired optical transmittance medium <b>620</b> or <b>640</b>.

<figref idrefs="DRAWINGS">FIG. 7</figref> shows another exemplary of an optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b> using only one PBS cube. At input, the PM fiber under test is connected to a PM collimator <b>701</b>, then light is coupled into a free-space PBS (<b>710</b>) from input port <b>711</b>. The collimator is aligned to make polarization P<b>1</b> and P<b>2</b> parallel to s-pol axis or p-pol axis of PBS, respectively. The PBS <b>710</b> splits the input light into S-polarized light at the output port <b>712</b> and P-polarized light at the output port <b>713</b>. The S-pol. light passes a polarization element <b>721</b> (e.g., a λ/4 wave plate or 45° Faraday rotator) and is reflected back to the port <b>712</b> of PBS <b>710</b> by a reflector <b>720</b>. At port <b>713</b> of PBS (<b>710</b>), the p-pol light is coupled into a single-mode (SM) fiber collimator <b>731</b> which is coupled to a SM fiber loop <b>732</b>. A fiber pigtailed Faraday mirror <b>733</b> is coupled to the fiber loop <b>732</b> to reflect the light back to the port <b>713</b> of PBS (<b>710</b>) by passing through the SM fiber loop <b>732</b> and the SM collimator <b>731</b>. At the output port <b>714</b> of PBS (<b>710</b>), s-pol. and p-pol light are combined by the PBS <b>710</b> into a combined light beam. A right-angle prism (<b>740</b>) is used to receive the combined light beam from the output port <b>714</b> of the PBS <b>710</b> and directed to the combined light beam into a PM collimator or a SM fiber collimator <b>750</b>. A 45° aligned polarizer <b>741</b> is provided between the SM fiber collimator <b>750</b> and the right-angle prism <b>740</b> to mix the S and P polarized light in the combined light beam. The delay between P<b>1</b> and P<b>2</b> at output <b>760</b> can be adjusted by changing the length of the fiber loop <b>732</b> via a fiber stretcher coupled to the fiber loop <b>732</b>.

<figref idrefs="DRAWINGS">FIG. 8</figref> gives an exemplary all-fiber optical delay device for generating a desired delay between two linear orthogonal polarization states P<b>1</b> and P<b>2</b> using one four-port pigtailed PBS. The output <b>801</b> of the PM fiber under test is connected into the PM input <b>811</b> of a 4-port fiber pigtailed PBS (<b>810</b>), the slow axis the PM fiber under test is aligned to the slow axis of input PM fiber of PBS, thus polarization states P<b>1</b> and P<b>2</b> are separated into two optical beams at two ports <b>812</b> and <b>813</b> of PBS <b>810</b>. Two SM/PM fiber loops <b>820</b> and <b>821</b> are connected to the ports <b>812</b> and <b>813</b>. Two 45-degree Faraday mirrors <b>830</b> and <b>831</b> are coupled to the terminals of the fiber loops <b>820</b> and <b>831</b>, respectively, to reflect the light signals in the two fiber loops <b>820</b> and <b>831</b> with a 90-degree rotation in polarization back to the PBS <b>810</b>. At the PM output <b>814</b> of PBS (<b>810</b>), the s-pol. and p-pol light reflected back in the fiber loops <b>820</b> and <b>821</b> are combined into one optical beam. The delay between the two polarization states P<b>1</b> and P<b>2</b> at output <b>850</b> can be adjusted by changing the relative length between fiber loops <b>820</b> and <b>821</b> by using one or two fiber stretchers.

<figref idrefs="DRAWINGS">FIG. 9</figref> further shows an exemplary device for measuring the distribution of polarization crosstalk of PM fiber based on the design in <figref idrefs="DRAWINGS">FIG. 2A</figref> and the optical delay device in <figref idrefs="DRAWINGS">FIG. 6A</figref>. A broadband light enters the PM fiber at position <b>901</b>, it has only one polarization component, aligned to the slow axis of PM fiber <b>910</b> under test. After passing through the PM fiber under test, the wave packet at input <b>901</b> is split to two group wave packets sequence along fast axis and slow axis respectively. These two wave sequences are aligned and input the delay device <b>600</b> in <figref idrefs="DRAWINGS">FIG. 6</figref>, at the output of <b>600</b> an additional delay ΔL is added between polarization state P<b>1</b> and P<b>2</b>. After passing through a 45° aligned polarizer <b>936</b> which mixes the two polarization states P<b>1</b> and P<b>2</b> at the output of the optical delay device <b>600</b>, the light is input into a Michelson interferometer <b>950</b> formed by an optical beam splitter PBS <b>951</b>, a fixed mirror <b>952</b> forming the reference arm <b>954</b> and a movable mirror <b>953</b> forming the adjustable optical arm <b>955</b>. The PBS <b>951</b> splits received light from the polarizer <b>939</b> into a first beam to the mirror <b>952</b> (e.g., by reflection) and a second beam to the mirror <b>953</b> (e.g., by transmission) and combines the returned light from both arms to produce a combined optical output. An optical detector <b>956</b> is used to receive the combined optical output from the PBS <b>951</b> to detect the optical interference information in the combined optical output. The delay between the reference arm <b>954</b> and moving arm <b>955</b> can be set by an offset ΔL generated by the delay device <b>600</b>. The interference pattern can be generated by moving the mirror <b>953</b> from zero to −Δn*l where l is the length of PM fiber under test. Using this setup, the zero-order interference will be suppressed and most of ghost peaks generated by the interference between multiple coupling can be suppressed or eliminated.

<figref idrefs="DRAWINGS">FIG. 10</figref> shows an exemplary device <b>1000</b> for measuring the distribution of polarization crosstalk of PM fiber using fiber pigtailed components based on the design principle in <figref idrefs="DRAWINGS">FIG. 2A</figref>. A linearly polarized broadband light source <b>1001</b> is used to generate input light to enter the PM fiber at input <b>1011</b> of PM fiber <b>1010</b> under test, where the optical polarization of the input light is aligned to the slow axis of PM fiber <b>1010</b> under test. After passing through the PM fiber <b>1010</b> under test, the wave packet at input <b>1011</b> is split to two groups of wave packet sequences which are polarized along the fast axis and the slow axis, respectively. These two wave sequences are input to the optical delay device <b>1030</b> which causes, at the output of <b>1030</b>, an additional delay ΔL(&gt;Δn*l where l is the length of PM fiber under test) between two orthogonal polarization states P<b>1</b> and P<b>2</b> of the PM fiber <b>1010</b> under test. A 45° aligned polarizer <b>1040</b> is placed downstream from the delay device <b>1030</b>, these two wave packet sequences are mixed with the same polarization state as the transmission axis of the polarizer <b>1040</b>. The light output by the polarizer <b>1040</b> is input to a fiber-based optical interferometer for optical interference measurements.

This fiber-based optical interferometer is constructed as shown by the box named “fiber-based optical interferometer.” The output of the polarizer <b>1040</b> is coupled to the port <b>1</b> of an optical circulator <b>1050</b> which directs the light from input port <b>1</b> to output port <b>2</b> of the circulator <b>1050</b>. A 4-port fiber coupler <b>1060</b> is coupled to the port <b>2</b> at the input port <b>1061</b> of the fiber coupler <b>1060</b>. The light received from the port <b>2</b> is split into separate signals at two output ports <b>1063</b> and <b>1064</b> of the fiber coupler <b>1060</b>. A variable delay line <b>1070</b> is coupled to the fiber port <b>1064</b> and a Faraday mirror <b>1081</b> is coupled to the other end of the optical delay line <b>1070</b> to reflect light back so that the polarization of the reflected light is orthogonal the polarization of the light incident to the Faraday mirror <b>1081</b>. A second Faraday mirror <b>1080</b> is coupled to the fiber output port <b>1063</b> of the fiber coupler <b>1060</b> to reflect the light back so that the polarization of the reflected light is orthogonal the polarization of the light incident to the Faraday mirror <b>1080</b>. The delay between the reference arm <b>1063</b> and the moving arm <b>1064</b> of the interferometer is set by an offset ΔL which is same as the delay generated by the delay device <b>1030</b>. The reflected light signals received at the ports <b>1063</b> and <b>1064</b> are spatially overlapped in the fiber coupler <b>1060</b> to produce optical interference which is contained in two output interference signals at the ports <b>1061</b> and <b>1062</b> of the fiber coupler <b>1060</b>. The output interference signal at the port <b>1061</b> is directed to the port <b>2</b> of the optical circulator <b>1050</b> which directs the signal to the port <b>3</b> as the first interference signal <b>1091</b>. The output interference signal at the port <b>1062</b> is directed as the second interference signal <b>1092</b>.

The interference signals <b>1091</b> and <b>1092</b> can be directed into a balance detector <b>1090</b> by changing the delay of the variable delay line <b>1070</b> from zero to −Δn*l where Δn and l are birefringence and the length of PM fiber under test. Using this setup, the zero-order interference is suppressed and most of ghost peaks generated by the interference between multiple couplings are eliminated during scanning of the delay line <b>1070</b> in the interferometer. A processing device <b>1100</b> is provided to receive the output of the balanced detector <b>1090</b>, which includes two optical detectors that receives the two optical outputs from the fiber-based optical interferometer: a first optical output from the fiber port <b>1061</b> of the fiber coupler via the optical circulator <b>1050</b> and a second optical output from the fiber port <b>1062</b> of the fiber coupler <b>1060</b>. The processing device <b>1100</b> processes the data in the output of the balanced detector <b>1090</b> to generate the desired measurements for locations of coupling points in the PM fiber and the magnitudes of the coupling at the respective coupling points.

<figref idrefs="DRAWINGS">FIG. 11</figref> shows an example of a device for measuring the attenuation ratio between two orthogonal polarization modes of light guided in a waveguide device, e.g., transverse electric (TE) mode and transverse magnetic (TM) mode. In this device, a linearly polarized broadband light source <b>1101</b> is used to generate input light to enter the input fiber <b>1111</b> of a waveguide device <b>1110</b> under test, where the optical polarization of the input light to the waveguide device <b>1110</b> is aligned to make TE and TM modes of the guided light in the waveguide of the waveguide device <b>1110</b> have equal power at the input of the waveguide device <b>1110</b>. After passing through the waveguide device <b>1110</b> under test, the wave packet received at input <b>1111</b> is split to two wave packet sequences which are polarized along ordinary optical axis (n₀, TM mode) and extraordinary optical (n_eTE mode) axis, respectively. These two wave packet sequences are input to the delay device <b>1130</b> which introduces, at the output of the delay device <b>1130</b>, an additional delay ΔL (greater than the delay generated by the waveguide device <b>1110</b> under test) added between two orthogonal polarization states TE and TM modes of the waveguide device <b>1110</b> under test. A 45° aligned polarizer <b>1140</b> is placed downstream from the delay device <b>1130</b> to cause these two wave packets to mix along the polarization direction of the transmission axis of the polarizer <b>1040</b>. This mix of the TM and TE modes at the polarizer <b>1140</b> causes optical interference. The light output by the polarizer <b>1140</b> is input to a fiber-based optical interferometer.

This fiber-based optical interferometer is constructed as shown by the box named “fiber-based optical interferometer.” The output of the polarizer <b>1140</b> is coupled to the port <b>1</b> of an optical circulator <b>1150</b> which directs the light from input port <b>1</b> to output port <b>2</b> of the circulator <b>1150</b>. A 4-port fiber coupler <b>1160</b> is coupled to the port <b>2</b> at the input port <b>1161</b> of the fiber coupler <b>1160</b>. The light received from the port <b>2</b> is split into separate signals at two output ports <b>1163</b> and <b>1164</b> of the fiber coupler <b>1160</b>. A variable delay line <b>1170</b> is coupled to the fiber port <b>1164</b> and a Faraday mirror <b>1181</b> is coupled to the other end of the optical delay line <b>1170</b> to reflect light back. A second Faraday mirror <b>1180</b> is coupled to the fiber output port <b>1163</b> of the fiber coupler <b>1160</b> to reflect the light back. The delay between reference arm <b>1163</b> and moving arm <b>1164</b> has already been set an offset ΔL which is same as the delay generated by delay device <b>1130</b>. The reflected light signals received at the ports <b>1163</b> and <b>1164</b> are spatially overlapped in the fiber coupler <b>1160</b> to produce optical inference which is contained at two output interference signals at the ports <b>1161</b> and <b>1162</b> of the fiber coupler <b>1160</b>. The output interference signal at the port <b>1161</b> is directed to the port <b>2</b> of the optical circulator <b>1150</b> which directs the signal to the port <b>3</b> as the first interference signal <b>1191</b>. The output interference signal at the port <b>1162</b> is directed as the second interference signal <b>1192</b>.

The interference signals <b>1191</b> and <b>1192</b> can be directed into a balance detector <b>1010</b> by changing the delay of the variable delay line <b>1170</b> from zero to −Δn*l where Δn and l are birefringence and the length of waveguide under test. Using this setup, the zero-order interference is suppressed and most of ghost peaks generated by the interference between multiple couplings are eliminated during scanning of the delay line <b>1170</b> in the interferometer. A processing device <b>1195</b> is provided to receive the output of the balanced detector <b>1190</b> and processes the data in the output to generate the measurements of the attenuation ratio between TM and TE modes of light guided in the waveguide device <b>1110</b> under test.

Based on the above described techniques and devices, a distributed polarization crosstalk analyzer can be constructed in a way to suppress or eliminate the undesired zero-order interference and to reduce the multi-coupling interference common in other interferometer devices for measuring distributed polarization crosstalk in PM fiber. In some implementations, a distributed polarization crosstalk analyzer based on present techniques and device designs can be configured as a sensing device by using the PM fiber itself as the sensing medium, eliminating the need to place multiple fiber gratings along the fiber in some grating-based sensing devices and obtaining higher spatial resolution of the stress distribution than sine grating-based sensing devices. Such a PM-fiber based sensing device has no discrete sensing elements and thus can be easy to install and calibrate, making it useful for monitoring space-resolved structural changes along bridges, tunnels, dams, oil pipes, or buildings. A distributed polarization crosstalk analyzer based on present techniques and device designs can be configured as an intrusion detection system to use mechanical disturbances to the PM fiber and the associated polarization coupling caused by an intrusion to detect the location of the intrusion. The present techniques and devices can also be used for PM fiber quality inspection for identifying defective sections of PM fiber and as a tool for detecting locations of imperfections or areas of local stress on the fiber coil induced during the fiber winding process. The software installed in the processing device in <figref idrefs="DRAWINGS">FIGS. 2A</figref>, <b>10</b> and <b>11</b> can be used to display the location and polarization coupling ratio of each stress point as a function of distance, and flags stress points with polarization coupling above a user defined trigger threshold. A quality inspection report window indicates the pass/fail status of the PM fiber or PM fiber coil, with a list of the locations and polarization coupling strengths of all stress points. Other applications of the present techniques and device designs include measuring high polarization extinction ratios of a polarizing waveguide, obtaining the autocorrelation function of a light source, measuring the birefringence of a PM fiber and the lengths of PM and SM fibers, and matching the optical path lengths of an interferometer.

<figref idrefs="DRAWINGS">FIG. 12</figref> shows an example of a process for measuring distributed polarization crosstalk in an optical birefringent medium such as PM fiber. Step <b>1210</b> of this process couples a linearly polarized light of a broadband spectrum into an optical birefringent medium in a direction along which the optical birefringent medium supports two orthogonal polarization modes due to optical birefringence to produce an optical output signal out of the optical birefringent medium. In Step <b>1220</b>, the optical output signal out of the optical birefringent medium is directed into an optical delay device which causes an added optical delay greater than an optical delay between the two orthogonal polarization modes caused by the optical birefringent medium in the optical output signal to produce a modified optical output signal. Step <b>1230</b> of this process directs the modified optical output signal to transmit through a linear optical polarizer which is polarized in a direction, e.g., 45 degrees with respect to one of the two orthogonal polarization modes of the optical birefringent medium, to cause a mixing between the two orthogonal polarization modes in optical transmission light of the linear optical polarizer. Step <b>1240</b> of the process directs the optical transmission light of the linear optical polarizer into an optical interferometer to obtain optical interference of light between the two orthogonal polarization modes in the optical birefringent medium. At Step <b>1250</b>, the obtained optical interference is processed to identify locations of the polarization coupling between the two orthogonal polarization modes in the optical birefringent medium. In addition, the obtained optical interference can be processed to obtain magnitudes of the polarization coupling at the identified locations.

The above described examples, the all-fiber device designs in <figref idrefs="DRAWINGS">FIGS. 10 and 11</figref> can be advantageous in various applications, even without the delay device <b>1130</b> described above. The all-fiber device designs can make the device compact in size, light in weight, and robust in performance. When such an all-fiber device in <figref idrefs="DRAWINGS">FIG. 10</figref> or <b>11</b> is implemented, the optical delay device <b>1130</b> can also be implemented based on all fiber designs such as the examples in <figref idrefs="DRAWINGS">FIGS. 5 and 8</figref>. Other fiber designs can also be used.

While this document contains many specific implementation details, these should not be construed as limitations on the scope of the invention or of what may be claimed, but rather as descriptions of features specific to particular embodiments of the invention. Certain features that are described in this document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, the separation of various system components in the embodiments described above should not be understood as requiring such separation in all embodiments.

Thus, particular embodiments of the invention and their implementations are disclosed. Variations, modifications and enhancements of the disclosed embodiments and implementations and other embodiments and implementations can be made based on what is described and illustrated in this document.